Investigation on natural frequency of an optimized elliptical container using real-coded genetic algorithm

11(2014) 113 - 129

Abstract
This study introduces a method based on real-coded genetic algo-
rithm to design an elliptical shaped fuel tank. This method enhan-
ces the advantage of the system such as roll stability, and reduces
disadvantages like fluid c.g. height and overturning moment.
These parameters corresponding to the elliptical tanks with diffe-
rent filling levels are properly optimized. Moreover the effects of
these optimized shapes on natural sloshing frequency are investi-
gated. Comparing presented results with experimental ones indica-
te the reliability and accuracy of the present work. In addition, a
new method based on genetic algorithm, which enhances tank
rollover threshold, is presented. This optimization enhances roll
stability, although reducing the natural sloshing frequency in
comparison to cylindrical tanks. In contrast, the sloshing frequen-
cy of the optimized elliptical tank is enhanced in compare with
conventional elliptical tanks, which is considered as an advanta-
geof the presented work.
Keywords
Genetic algorithm, sloshing, optimization, tank vehicle, overtur-
ning moment.
Investigation on natural frequency of an optimized
elliptical container using real-coded genetic algorithm
1 INTRODUCTION
Generally, Rollover accidents are one of the most common types of accident that occur in com-
mercial vehicles carrying fluid. Since most of the time, these tankers carry dangerous liquids such
as ammonia, gasoline, and fuel oils, therefore stability of partially filled liquid cargo vehicles are
of great importance (Acarman, 2006). Sloshing is a fluid oscillation phenomenon caused by the
tank motion. Fluid oscillation depends on the tank geometry, filling conditions, and frequency
range inside the tanker (Budiansky, 1960). Besides, sloshing frequency and magnitude of sloshing
forces are also dependent on these parameters (Hasheminejad and Aghabeig, 2011). If the sloshing
frequency is close enough to the structural natural frequency, resonance will occur (Ibrahim,
1975). In addition, the stability is one of the most important points in design of vehicles used for
carrying and storing objects and substances. There are many researches devoting on sloshing fre-
M. H. Shojaeefard
a
R. Talebitooti
a
S. Yarmohammadi Satri
a, *
Mohammad Hassan Amiryoon
b
a
School of Mechanical Engineering., Iran
University of Science and Technology, Teh-
ran, Iran.
b
Center of Excellence for Power System
Automation and Operation, Department of
Electrical Engineering,Iran University of
Science and Technology, Tehran, Iran
Received in 29 Dec 2012
In revised form 09 Apr 2013
*
Author email: yysadegh@yahoo.com

114 S. Y. Satri et al./ Investigation on natural frequency of an optimized elliptical container using real-coded genetic algorithm
Latin American Journal of Solids and Structures 11(2014) 113 - 129
quency of tankers. Nonlinear dynamic response of a system which is subjected to harmonic exter-
nal force, have been studied by interacting lowest fluid sloshing mode with the various modes of
structure. Internal resonance and combined resonance profile are presented as results (Ibrahim,
1988). Nakagawa and Ikeda (1997) studied theoretically and experimentally non linear vibrations
of a two-dimensional rectangular structure and water sloshing. Aliabadi et al. (2003)conducted a
comparison between the numerical analysis, fluid mechanics, and mechanical models. Navier-
Stokes equations have been solved with the aid of finite element method to measure accuracy of
pendulum model under constant lateral acceleration. Based on these results, both methods valida-
te each other under low fuel tank filling. The difference between ranges of frequencies in these
two methods is considerable when there is a large amount of fuel in the tank. Mechanical models
such as roll plane model have been developed as an alternative approach to the study of fluid
sloshing. The roll plane model is determined by center of gravity, resulting force and torque, fre-
quency of oscillation, and mass and inertia characteristics of the mechanical system, which is
equal to tank fluid slosh (Li and Wang, 2012). The roll stability of a partially filled vehicle is
influenced by both the c.g. height and the magnitude of lateral load transfer. Different cross-
sections have been analyzed to optimize the roll stability. The circular cross section has a high
center of fluid location, but considerably less lateral load transfer with low fill volumes under a
steady turning lateral acceleration field. On the other hand, modified oval tanks when compared
to circular tanks yield lower c.g. height and relatively larger lateral load transfer with low fill
volumes. Therefore, the modified oval tanks represent higher roll stability limits than circular
tanks (Kang et al., 1999; Rakheja et al., 1988; Rakheja, 2002; Hyun-Soo, 2008). In order to opti-
mize the elliptical shape of road containers, a numerical analysis is developed which minimizes
overturning moment according to the position of fluid c.g. (Popov, 1996). The results indicate
that the optimal ratio of height and width in elliptical tankers has decreased in terms of lateral
momentum.Mostresearches in the field ofsustainabletanker concentrate on enhancing roll stability
with the usage of baffles. Optimizing cross-section and effects of natural frequency on optimized
tanks have not been considered in recent studies.
In this article, the elliptical tank shape is initially optimized based on real value method of ge-
netic algorithm. Then, overturning moment and fluid c.g. height are minimized simultaneously.
The obtained cross-sections enhance roll stability in different filling levels. The influence of GA
parameters such as mutation rate and population on the accuracy and rapidity of results conver-
gence are properly described. Moreover, the natural frequency of optimized shape, and also the
effects of optimization on sloshing frequency are discussed. Finally the results of the presented
research are compared with those of conventional circular, elliptical, and generic tank cross-
sections.
2 QUASI-STATIC EQUILIBRIUM FORMULATION
Assuming the quasi-static fluid sloshing, the roll plane model of a tanker is established. Lateralac-
celerationandroll of the tankercausesmoving the fluidmasscenterfrom pointto pointcl to the point
xl . Inthis study, therotationangleisassumed to be small andviscousfluid is Negligible. Eq. (1)shows
theshape ofanellipticaltanker.

S. Y. Satri et al./ Investigation on natural frequency of an optimized elliptical container using real-coded genetic algorithm 115
x2
a2
+
y − b( )2
b2
=1 (1)
where x and y are the coordinates of elliptical geometry, 2a and 2b are elliptical height and width
respectively, Eq. (1) can be rewritten as:
y = b 1± 1−
x
a
⎛
⎝⎜
⎞
⎠⎟
2⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
(2)
To obtain themass andcenter of gravity,equationswhichrepresentsthe fluidmotion,should beu-
sedrepeatedly.The equation offluid free surface withoutsprung mass roll angleand lateral accelera-
tion represented as:
y = −h0
(i)
, i =1,...,n (3)
while Equations (1-3) are solved simultaneously the intersecting points of the fluid free surface
containing (x1,y1) and (x2,y2) are obtained (Figure 1). After comparing h0
(i)
with themaximum and
minimumvalues ofthe arcs limits,cross-sectional areaare obtained.
Since(x1,y1) and (x2,y2)are symmetricalpoints with respect toyaxis in theabsence oflateral accelera-
tion and roll angle of sprung mass, therefore x1=-x2 and volumeoffluidper unit of tank lengthis
predictedasfollows:
A0
(i)
= 2 dxdy
f1(x)
h0 (i)
∫0
yp
∫ (4)
f1(x) expresses the equationoftank geometry. The equation offree fluid surface under steady tur-
ning presented as:
y = −
ϕS − ay
1+ϕSay
⎛
⎝
⎜
⎞
⎠
⎟ x +α(i)
,i =1,...,n (5)
ϕs
is the roll angle of sprung mass, ay is the lateral acceleration in g (gravity acceleration) unit
and α(i)
is intersection of the y-axis withthe free surface ofthe fluid.

2 2( , )x y
1 1( , )x y
θ
cgx
cgy
lx
lc
Figure 1 Elliptical parameters under steady lateral acceleration
The totalvolume offluidper unit lengthis compared with theinitial volumeinthe case that ay = 0
and ϕs = 0 . The error values corresponding to fluid volume are estimated as:
ε =| A(i)
− A0
(i)
| (6)
Theprocess would be repeated again until the error function rate goes beyond the allowable
error.
Yl
(i)
=
1
A(i)
ydydx
f1(i)
f2
(i)
∫
yp
yq
∫ ;i =1,2,...,10
Xl
(i)
=
1
A(i)
xdydx
f1(i)
f2
(i)
∫
yp
yq
∫ ;i =1,2,...,10
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
(7)
For one iteration if b ≤ h ≤ 2b , cross-sectional area (Ah) and fluid c.g. height (ycg) are represented
as:
Ah = πab−α (8)
α = 2 xhb − xhh +
b
2a
xh a2
− xh
2
+ a2
sin−1
(
xh
a
)
⎡
⎣⎢
⎤
⎦⎥
⎧
⎨
⎩
⎫
⎬
⎭
(9)
yCG =
πab2
−αβ
Ah
(10)
while
β = 2b − yCG (11)

3 TANK VEHICLE ROLL PLANE MODEL
In order to optimize the tank cross-section, c.g. height and overturning moment are in challenge
to each other. In low filling, circular cross-section presents more stability, as its overturning mo-
ment is less than elliptical tankers. The fluid c.g. height is dominated in high filling volumes.
Therefore an elliptical tanker which has a lower center of gravity, represents more stability. In
addition lateral and vertical liquid load shifts affect vehicle roll stability. In other handliquid load
shift reduces as vehicle roll stability increases. This happens by minimizing overturning moment
due to lateral acceleration and fluid c.g. height. The overturning moment is calculated with the
geometric variables such as elliptical height and width (that is employed for computing fluid
weight (W), vertical and horizontal component of fluid c.g. (Xcg, Ycg)) (Romero, 2005; Romero,
2007). In order to perform the optimization process, a vehicle model based on the one presented
in Figure 2 with characteristics listed in Table 1 is employed. The fluid overturning moment is
obtained by solving dynamical equations which are shown in the following.
mszs = Fs,k,o + Fs,c,o + Fs,c,i + Fs,k,i − msg
Is
θs = (Fs,k,o − Fs,k,i )s + (Fs,c,o − Fs,c,i )s + ms (g − ys ) θs (hcs − hrs )[ ]+ Mz
⎧
⎨
⎪
⎩⎪
(12)
mu
yu = Ft,k,o + Ft,c,o + Ft,c,i + Ft,k,i − (Fs,k,o + Fs,c,o + Fs,c,i + Fs,k,i ) − mug
Iu
θu = (Ft,k,o − Ft,k,i )b + (Ft,c,o − Ft,c,i )b − (Fs,k,o − Fs,k,i )s − (Fs,c,o − Fs,c,i )s
+ mu(g − yu ) θu(hca − hra )⎡⎣ ⎤⎦ + mu(aL − yL)(hcu − hru )
⎧
⎨
⎪
⎪
⎩
⎪
⎪
(13)
For the sprung mass following equations can be written as:
Fs,k,o = ks(ys + sθs )
Fs,c,o = cs( ys + s θs )
Fs,k,i = ks(ys − sθs )
Fs,c,i = cs( ys − s θs )
(14)
For the unsprung mass following equations can be written as:
Ft,k,o = kt (yu + bθu)
Ft,c,o = ct ( yu + b θu)
Ft,k,i = kt (yu − bθu)
Ft,c,i = ct ( yu − b θu)
(15)
M0 = W (Ycg cosϕ − Xcg sinϕ )a +Ycg sinϕ + X cg cosϕ⎡⎣ ⎤⎦ (16)

b
c
b
a
ϕ β
θsi
si
sj
a
lW
ap
r
ui
uj
i
b i
s
1iFT 2iFT
3iFT 4iFT
id
iR
1i
FS
2i
FS
caih
rsih
θui
ri
z
ui
z
o
, ,S k o
F
, ,S c o
F
, ,S k i
F
, ,S c i
F
csih
Figure 2 Roll plane model of elliptical tankers
Table 1 Specification of tanker used in roll plane model
Tank truck characteristics
ms, sprung mass 5000 kg
mu, unsprung mass 1000 kg
mc, cargo mass 1000 kg
ks, suspension stiffness 1000 kN/m
cs, suspension damping 15 kNs/m
kt, dual tire stiffness 1400 kN/m
ct dual tire damping 2.5 kNs/m
b, half track width 1.0 m
s, half spring spacing 0.7 m
hcs sprung mass c.g. height 1.5 m
hca, axle c.g. height 0.6 m
hrs, sprung mass roll center height 1.0 m
R, turning radius 250 m
where ms, mu and mc are sprung mass, un-sprung mass and cargo mass. ks is suspension stiffness,
cs suspension damping, kt dual tire stiffness, ct dual tire damping, b half track width, s half spring
spacing, hcs c.g. height of sprung mass, hrs height of sprung mass roll center. Fs,k,o and Fs,k,i are
inner and outer spring forces in sprung mass. Fs,c,oand Fs,c,iare inner and outer damping forces in

unsprung mass. Ft,k,oand Ft,k,i are inner and outer spring forces in unsprung mass. Ft,c,oand
Ft,c,iare inner and outer damping forces in unsprung mass. sθ and uθ are roll angle of sprung mass
and unsprung mass. The gradient of the liquid free surface is computed as Eq (17).
ϕ =
A +θs
1+ Aθs
(17)
In order to verify the presented roll plane model, an elliptical tank with the tank width of 1.6
m, height of 0.8 m, tank compartment length of 2.5 m and cross sectional area of 4 m2
is consid-
ered. These characteristics are applied based on dimensions of Popov (1996) tank. The results of
overturning moment for this particular shape are compared with those represented by Popov
(1996). As depicted in the Figure 3 the excellent agreements are achieved which shows the relia-
bility of the model.
Figure 3 Validation of tank vehicle roll plane model based on overturning moment
4 NATURAL SLOSHING FREQUENCY
Fluid natural frequency should not be located near the tank resonance range in tank maneuvers.
Natural sloshing frequency increases with enhancement of roll stability and fluid c.g. height in
elliptical and circular tankers (Romero, 2007).Romero (2005) used two experimental and theoreti-
cal methods to estimate sloshing frequencies. The experimental method is based on effects of tank
filling level and tank shape on the sloshing frequency under c.g. fluid shift. In general, longitudi-
nal sloshing frequency is greater than the lateral sloshing frequency. This means that lateral natu-
ral frequency is more important than the longitudinal. In other word, the resonance phenomena
occurs in lateral mode. In the theoretical method, the wave velocity equation is obtained using
some simplifications including infinite fluid depth, and fixed and variable flow. In an ideal incom-
pressible liquid, fluid surface velocity in rotating mode is a function of wave length λ , tank depth
0.6 0.7 0.8 0.9 1
4
4.5
5
5.5
6
6.5
7
7.5
8
x 10
5
Filling Level
OverturningMoment(N.m)
Tank vehicle model
Popov(1999)

h, and gravity acceleration g. It is independent of other features such as fluid density. In general,
the natural sloshing frequency in a rectangular tank is expressed as follows:
c = λ f =
g
k
tanhkh
⎛
⎝⎜
⎞
⎠⎟
1
2
(18)
The resonance frequency in slow variation of speed is presented as:
f =
c
2L
(19)
where L is the wave length and c indicates the average wave speed and it is computed by the
Eq. (19), as follows:
c =
L
2
dx
c(x)0
L
2
∫
(20)
The effect of wave length or in other word tank width especially for elliptical tank, in the case
of fast wave speed variation should be mentioned in Eq. (21). The wave speed diminishes as a
result of rapid depth difference, in both ends of elliptical surface. Therefore effective wave length
(Lc) is actually shorter than the value measured in shape (L). Eq. (18) is used for determining
sloshing frequency when the wave speed variation is in a limited band as Eq. (20).
d

c(x)
dx
≥ acrit (21)
d

c(±
Lc
2
)
dx
≥ acrit (22)
Comparing the results of both experimental and theoretical methods indicates the deviation of
10% approximation. Natural sloshing frequencies for different shaped containers are obtained
through substituting Eq. (23) (heq) with h in Eq. (18) which is shown in Figure 4 The equivalent
fluid depth heq is determined as a function of the liquid cross-sectional area and the fetch length
L.
heq =
Aarea
L
(23)

2a
2b
L
λ
L
Figure 4 Equivalent fluid depth for a non-rectangular container
5 GENETIC ALGORITHM
The evolution algorithmspresent optimization techniques that are used in a wide range of opera-
tions such as parametric optimization, research, automatic generation of computer programs, and
combined issues (Salehi and Tavakkoli-Moghaddam, 2009; Corriveau et al., 2001). Contrary to
the traditional optimization techniques which use derivation in optimization evolutionary algo-
rithms and utilize fitness function, evolutionary algorithms represent a set of individuals on opti-
mization. The search region is divided into subsets that make it possible to achieve the general
optimal point. The natural evolution operates by means of three main processes of evolution,
mutation and selection. The main object of the mutation is producing new population and main-
taining variation in the natural population. Effects of mutation rate selection on quickness and
accuracy of fitness function convergence are studied in this paper. The crossover operator is used
for transmitting good properties of the two populations as parents to a new population called
offspring (Figure 5).
2 1.5 3.1 4.2 5.1 6.3 4.6 3.1 2.5 1.2 3.4 5.3
3.2 4.1 2.4 6.2 1.4 2.2 4.4 8.1 6.7 8.3 9.4 2.8
Parents
3.2 1.5 2.4 6.2 5.1 6.3 4.4 8.1 6.7 1.2 3.4 5.3
2 4.1 3.1 4.2 1.4 2.2 4.6 3.1 2.5 8.3 9.4 2.8
offsprings
Cross over
2 1.5 3.1 4.2 5.1 6.3 4.6 3.1 2.5 1.2 3.4 5.3Parent
2 0.5 0.8 4.2 1 6.3 4.6 0.1 0.7 0.9 3.4 5.3offspring
Mutation
Figure 5 Real-coded genetic algorithm operators

Scattered crossover operator is used for minimizing overturning moment. Just like nature, su-
perior populations have more election possibility and effects on their perimeter. To model the
process of evolution in the algorithm, fitness function is defined as a combination of overturning
moment and fluid c.g. height. Using this application, the population with greater fitness, has
more effect on the future. In generation t, evolutionary algorithm consists of p chromosomes in
the population. In t=1, the number of N initial population is randomly generated. The produced
target populations are decoded, and for each of them correspondingly a fitness function is consid-
ered. In order to solve optimization problems different methods such as real-coded and bit string
is utilized. The GA real-coded technique is applied in the present optimization problem. There-
fore, the calculations are based on real value of variables. The Figure 6 represents a diagram that
investigate elliptical cross-section that give more stability to tank vehicle. Some constraints such
as cross-sectional area and bounds of variables lead into the minimization. The variables of a
andb in this diagram equal to the width and height values. This optimization determines two
variables as fluid c.g. height (Ycg) and overturning moment (M0) in order to calculate the objec-
tive function as:
U (X ) = Minimize [w1
M0
+ w2
YCG
] (24)
The algorithm specifications such as theobjective function, constraints, design variables and their
lower and upperbounds are briefly mentioned in Table 2.The present method leads to selecting op-
timum variables and curving shapes that minimize overturning moment and fluid c.g. height.
Table 2 The Specifications of the genetic algorithm
The genetic algorithm characteristics
objective function U=w1M0+w2Ycg
tank variables
tank width (2b)
tank height (2a)
optimization non linear constraint (m
2
) Area ( π= ≤ 6.4A ab )
optimization linear constraint (m)
≤ ≤0.8 1.8a
≤ ≤1.1 2.1b

Genetic algorithm
Evaluating Fitness
Termination
Criterion Test?
Selection/Elitism
Crossover
Mutation
Best rate:4%-6%
Multi-objective Function
Determination
( sum of weighted cost functions)
Determining
Optimized Variables
Deriving optimized
Cross-section
Calculating Fitness
and Frequency
Assign Weights to
Each Objective
Function
Multi-objective Weighting
Algorithm
Training Data
Frequency Analysing
Define feasible solution
space
Initial Population
(comparing optimized on
with conventional)
No
Yes
Figure 6 Flowchart of optimization using GA
The present multi-objective optimization method is converted to a single objective function
based on sum of weighted cost function method. The weighting coefficients are chosen according
to their importance (Kaur and Bakhshi, 2010; Haupt, 2004). The approach of Assign Weight To
Each Objectives is used to determine different values assumptions for each sub-population on the
relative worth of cost functions. The two vector sizes are determined in the case of a bi-criteria
research (Pei-Chann and Chen, 2009). The weighted vector and cost functions are presented as
follows:
F = wm1 f1 + wm2 f2, f1 = Ycg, f2 = M0 (25)

(wm1,wm2 ) = (
1
m
,1−
1
m
), m = 2,3.... (26)
where f1 and f2 are the objective functions in each generation and m is the mth stage of choosing
weighting coefficients. The Eq. (26) is obtained from studying the effects of objective functions on
roll stability, in this paper. This process is repeated until the correct answers have been achieved.
The obtained frequency and overturning moment are compared with the data of conventional
elliptical tank, in order to omit unsuitable weighting coefficients through the algorithm which is
shown in Figure 6.
5.1 Developing Genetic Algorithm
In order to find the best overturning moment that enhances roll stability of tankers a MATLAB
code matching genetic algorithm is developed. Then, tank sloshing frequency is analyzed to avoid
resonance phenomenon. In order to find the best elliptical tank shape with the highest rollover
threshold, both elliptic parameters indicated as a and b (elliptic width and height) are introduced
as unknown parameters, where the corresponding area is constrained. Moreover, the code has
ability to investigate different design variables such as tank cross-sectional area, oval diameter,
overturning moment, fluid c.g. height and sloshing frequency. The flowchart of genetic algorithm
demonstrated in Figure 6, can be applied to achieve and accurate reliable results with genetic
operators (mutation, cross over and etc). Individuals generated randomly in each generation, in-
dicate overturning moment and fluid c.g. height simultaneously. They have the possibility of
being out of feasible tank dimension. Therefore, to reduce the iteration number of generations and
prevent improper individuals in population, a conventional tank shape and also some constraints
are introduced for elliptical shape. Moreover, There are many points that have a low overturning
moment in domain but they are not appropriate to be as an elliptic tank shape. These points lead
into increasing the fluid c.g. height. In this respect, the objective function is determined as a
combination of fluid c.g. height and overturning moment.
5.2 Analyzing the Mutation Rate
In this section, the mutation rate, which determines accuracy and speed of convergence response
in the GA program, is investigated. The code developed here, includes a large number of explora-
tions for different items such as population number, iteration number, value of cost function, and
mutation rate. Thus, minimum value of iteration and speed of convergence are very important.
To explore the effect of mutation rate an elliptical tank shape with the area of 3.256 (m2
) is
taken. The mutation rates are taken the values of 0, 2%, 4%, 6%, and 10%. As presented in Figu-
re 7, without any mutation rate, an early convergence occurs and GA algorithm cannot satisfy all
the possible value of constraints. For the case of 2% mutation the individual bit changes and the
chance of achieving better fitness functions increases. While, the mutation equals zero, the best
fitness function is obtained at 53 generation and the minimum overturning moment will be 55000
N.m. However, while the value of mutation rate takes 2%, proper fitness function is obtained at
81 generation and the minimum overturning moment is 560000 N.m. As represented in Figure 7,

enhancement of the mutation rate value increases the probability of various individuals in the
population. Based on this explanation, the mutation rate should be selected between 4% and 6%.
Moreover an appropriate population number would affect on achieving the best overturning mo-
ment. To explore the effect of population number and mutation rate, two filling levels of 70% and
80% of an elliptical container are studied.
Figure 7 Comparing convergence of fitness function in different mutation
As illustrated in Figure 8, for all mutation rates except 0% and 10%, the results are well con-
verged. However, while the population is fewer than 40 or mutation rate is fewer than 2% and
over 10%, fitness functions that represent overturning moment and as a result frequency is not
compatible. Therefore, the population number should be at least 40.
The results of presented research based on real-coded GA are compared with other tank vehi-
cles. As shown in Figure 9overturning moment of differrent tank shapes such as circular tank and
conventional elliptical tank are compared with optimized elliptical tank shape. Results show that
optimized tank shape has minimum overturning moment compared to other tank shapes. In other
hand, using optimized GA real-value method increase tank stability, although it has a reverse
effect on sloshing frequency.
The natural sloshing frequencies of optimized elliptical tank are compared with those of con-
ventional elliptical and circular ones that are obtained based on numerical and analytical analyzes
(Figure 10). It is well observed that converting the cylindical shapes into elliptical one decreases
the natural frequency of the sloshing mode which is not suitable. It also depicts that the natural
frequency of optimized elliptical shape is somewhat higher than conventional ones.
In this study, two different filling conditions (70% and 80%) of optimal elliptical tank that is
obtained based on conventional circular forms MC307 and MC312 with a diameter of 2.03 mare
illustrated. After analyzing related optimizing results, it is concluded that optimal elliptical cross-
section for both filling conditions has the best convergence in 120 individual numbers. The opti-
mal cross-section is presented in Figure 11 As depicted in this figure the oval parameters a and b
obtained for instance in 90% filling level, the optimal oval diameters are equal to a=1.6 and
0 10 20 30 40 50
3
3.5
4
4.5
5
Generation
Fitnessvalue
Mutation 0%
Mutation 3%
Mutation 6%
Mutation 9%

b=1.28m, which they are optimized values of a circular with radius of 1.427m. The convergence is
obtained after 102 iterations.
0 20 40 60 80 100 120 140 160 180
0.6
0.8
1
1.2
1.4
1.6
Population size (N)
Fitness
mutation 10%
mutation 6%
mutation 4%
mutation 2%
mutation 0%
0 20 40 60 80 100 120 140 160 180
0.5
1
1.5
2
2.5
3
x 10
5
Population size (N)
Overturningmoment(N.m)
mutation 10%
mutation 6%
mutation 4%
mutation 2%
mutation 0%
0 20 40 60 80 100 120 140 160 180
0.3
0.35
0.4
0.45
0.5
0.55
Population size (N)
Frequency(Hz)
mutation 6%
mutation 10%
mutation 2%
mutation 4%
mutation 0%
0 20 40 60 80 100 120 140 160 180
0.5
0.6
0.7
0.8
0.9
1
1.1
Population size (N)
Fitness
mutation 10%
mutation 6%
mutation 4%
mutation 2%
mutation 0%
0 20 40 60 80 100 120 140 160 180
6
7
8
9
10
x 10
4
Populaation size (N)
Overturningmoment(N.m)
mutation 10%
mutation 6%
mutation 4%
mutation 2%
mutation 0%
0 20 40 60 80 100 120 140 160 180
0.3
0.35
0.4
0.45
Population size (N)
Frequency(Hz)
mutation 10%
mutation 4%
mutation 0%
mutation 2%
mutation 6%
80 Percent Filling Level70 Percent Filling Level
Figure 8 Fitness function, overturning moment and frequency in different population size and mutation rate

Figure 9 optimal elliptical overturning moment
Figure 10 Natural frequency in different shape
0.6 0.65 0.7 0.75 0.8 0.85 0.9
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
5
Filling level
OverturningMoment(N.m)
Optimized elliptical tank vehicle
Conventional elliptical tank vehicle (Popov (1999))
Circular tank vehicle (Hasheminejad (2011))
0.6 0.65 0.7 0.75 0.8 0.85 0.9
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Filling level
NaturalFrequency(Hz)
Elliptical (Romero (2005))
Cylindrical (Romero (2005))
Optimized elliptical
Oval (Romero (2005))

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0.7
0.8
0.9
1
1.1
1.2
X (m)
Y(m)
80%
70%
60%
90%
Figure 11 Optimized elliptical corss-sections in different filling conditions
6 CONCLUSION
In this paper, a method based on real-coded genetic algorithm was proposed to specify the best
elliptical cross-section with minimum fluid c.g. height and overturning moment in the feasible
tank cross-sectional area. In this reason, a code was developed to predict the optimized elliptical
cross-section. In addition natural sloshing frequency was considered as a factor of tank rollover.
The results presented in different filling conditions were compared with conventional methods.
Processing the results indicates that the present method is a suitable way to determine optimized
elliptical tank shape. The results also show that the natural sloshing frequency is enhanced in
comparison to conventional elliptical tanks. It is a worthwhile result, as it indicates the roll stabi-
lity of optimized tank in a stable margin.
The results also indicated that the population number should be at least 40 and mutation rate
is suggested 4% to 6%. Finally, it should be noted that the proposed model is a fast and accurate
method to design elliptical tank shape, which increases natural sloshing frequency in compare
with conventional elliptical tank cross-sections.

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